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Mirrors > Home > ILE Home > Th. List > seqeq1 | Unicode version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . 6 | |
2 | fveq2 5421 | . . . . . 6 | |
3 | 1, 2 | opeq12d 3713 | . . . . 5 |
4 | freceq2 6290 | . . . . 5 frec frec | |
5 | 3, 4 | syl 14 | . . . 4 frec frec |
6 | fveq2 5421 | . . . . . 6 | |
7 | eqid 2139 | . . . . . 6 | |
8 | mpoeq12 5831 | . . . . . 6 | |
9 | 6, 7, 8 | sylancl 409 | . . . . 5 |
10 | freceq1 6289 | . . . . 5 frec frec | |
11 | 9, 10 | syl 14 | . . . 4 frec frec |
12 | 5, 11 | eqtrd 2172 | . . 3 frec frec |
13 | 12 | rneqd 4768 | . 2 frec frec |
14 | df-seqfrec 10219 | . 2 frec | |
15 | df-seqfrec 10219 | . 2 frec | |
16 | 13, 14, 15 | 3eqtr4g 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 cvv 2686 cop 3530 crn 4540 cfv 5123 (class class class)co 5774 cmpo 5776 freccfrec 6287 c1 7621 caddc 7623 cuz 9326 cseq 10218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fv 5131 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-seqfrec 10219 |
This theorem is referenced by: seqeq1d 10224 seq3f1olemqsum 10273 seq3id 10281 seq3z 10284 iserex 11108 summodclem2 11151 summodc 11152 zsumdc 11153 isumsplit 11260 ntrivcvgap 11317 ntrivcvgap0 11318 prodmodclem2 11346 prodmodc 11347 ege2le3 11377 |
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